A Distributed PID-like Consensus Control for Discrete-time

Multi-agent Systems

Nicol

o Gionfra

, Guillaume Sandou

, Houria Siguerdidjane

and Damien Faille

Laboratoire des Signaux et Syst

emes (L2S), CentraleSup

elec, Universit

e Paris-Saclay, 3 rue Joliot Curie,

91192 Gif-sur-Yvette, France

EDF R&D, Department STEP, 6 quai Watier, 78401 Chatou, France

Keywords:

Multi-agent Systems, Consensus Control, Discrete-time LTI Systems.

Abstract:

The problem of discrete-time multi-agent systems governed by general MIMO dynamics is addressed. By

employing a PID-like distributed protocol, we aim to solve two relevant consensus problems, namely the lead-

erless consensus under disturbances and leader-follower under time-varying reference state ones. Sufﬁcient

conditions for stability as well as two LMI approaches to tune the controller gains are provided. The latter are

either based on a H

∞

formulation of the problem or on fast response to a reference exogenous signal. Numer-

ical simulations give some insight of which tuning should be considered according to the problem addressed.

1 INTRODUCTION

In recent years much research effort has been de-

voted to the area of multi-agent cooperative control

because of its wide range of applications and potential

beneﬁts. Cooperation of a coordinated multi-agent

network is sought via distributed algorithms as they

present some interesting advantages over their cen-

tralized counterpart, e.g. avoiding single point of fail-

ure, reducing communication and computational bur-

den, etc. The main problem in distributed coordi-

nation, known as consensus problem, is the one of

achieving an agreement on some variables of interest

of each agent via local interactions. These variables

evolve according to a prescribed dynamics describing

the physics of the problem, while interactions among

agents are deﬁned by a given communication graph.

Finding a distributed protocol to solve the aforemen-

tioned problem has been extensively treated for single

and double integrator dynamic agents, e.g. (Ren and

Beard, 2008). However, in a more general framework,

general dynamics need to be considered in order to

describe the agents behavior.

The consensus problem for this latter case has

been discussed for both continuous and discrete-time

multi-agent systems. In addition, it can be further di-

vided in two main classes of problems, namely lead-

erless and leader-follower ones. As far as the former

is concerned, the most employed distributed protocol

is given by a static state feedback law, also called P-

like distributed control. One can cite, for instance,

(Xi et al., 2010), (Li et al., 2013), (Yang-Zhou et al.,

2014) for the continuous-time framework, and (Li

et al., 2013), (You and Xie, 2011), (Su and Huang,

2012), (Ge et al., 2013) for the discrete one, where the

consensus problem is led back to the one of simulta-

neously stabilizing multiple LTI systems. References

(Li et al., 2013), and (Su and Huang, 2012) also solve

a leader-follower problem where the leader has an au-

tonomous time-invariant dynamics. Another interest-

ing problem is the one of ﬁnding the optimal P-like

protocol gain in order to improve consensus under

system uncertainties, as in (Li et al., 2012), and dis-

turbances as in (Oh et al., 2014), (Li et al., 2011), for

continuous time systems, and (Wang and Gao, 2011)

for discrete-time ones. The proposed approaches usu-

ally make use of some H

or H

∞

constraints to be re-

spected, and they are in general more involved than

the one of simultaneously stabilizing multiple sys-

tems. For instance, (Li et al., 2011) provide neces-

sary and sufﬁcient conditions, for the continuous-time

case to solve the consensus problem while guarantee-

ing some properties on the aforementioned norms. On

the other hand, for discrete-time systems only sufﬁ-

cient conditions are provided using results from ro-

bust control as in (Wang and Gao, 2011). Dynamic

distributed controllers are also proposed for consen-

sus achievement based on local output measurements,

e.g. (Li et al., 2013). In the continuous-time frame-

work, (Xi et al., 2012) provide a controller with lim-

ited energy, while a general full order one is presented

Gionfra, N., Sandou, G., Siguerdidjane, H. and Faille, D.

A Distributed PID-like Consensus Control for Discrete-time Multi-agent Systems.

DOI: 10.5220/0006420500720081

In Proceedings of the 14th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2017) - Volume 1, pages 72-81

ISBN: 978-989-758-263-9

in (Liu et al., 2009) to achieve some H

∞

performance.

Other possible structures have been explored too. In-

deed, given the common P-like controller, one can

easily think of a more general PID-like structure. In

continuous-time, for instance, (Carli et al., 2008) pro-

pose a PI-like distributed algorithm for single integra-

tor dynamic agents, and (Ou et al., 2014) provide a

PID-like controller for general high-order SISO sys-

tems. Similar control design is applied to solve a

leader-follower consensus under time-varying refer-

ence state, as in (Ren, 2007), and in its sampled-data

counterpart (Cao et al., 2009), where a PD-like proto-

col is given. Even though the presented literature re-

view is nowhere near exhaustive, one can remark that

poorer attention has been devoted to discrete-time dy-

namic protocols for general LTI MIMO systems, and

this is where we wish to place our contribution.

In this paper we propose a PID-like distributed

controller for the aforementioned systems, where the

agents can communicate on a connected undirected

graph, and we provide two possible ways of tuning

the controller parameters, based on the solution of

LMIs. To the best of our knowledge this distributed

control structure has never been fully treated for the

mentioned class of dynamic systems. The approach

we propose is used to solve two different problems,

namely the leaderless consensus under the presence

of disturbances, and the leader-follower consensus

under a time-varying reference state. Our main re-

sults are based on the work of (Wu et al., 2011),

which we adapted for distributed coordination pur-

poses. The fundamental feature of the aforesaid work

is that MIMO PID parameter tuning can be performed

via LMIs, avoiding in this way, the need for solving

BMIs. Furthermore, in both the analyzed consensus

problems the measurement matrix is kept general, al-

lowing a more general problem formulation for the

case in which the agents cannot directly measure the

variables on which agreement is sought. Eventually,

concerning the leaderless consensus, agreement can

be focused on particular variables of interest via a

proper selection of the controlled output matrix. As

for classic control, the PID controller allows good

performance despite being rather simple. Concerning

the leaderless consensus problem, for instance, it en-

hances the disturbance rejection, and achieves results

that a simple P-like protocol would not permit if the

dynamics of the agents are general. Similar conclu-

sions hold for the leader-follower consensus problem

with a time-varying reference state, where a P-like

control would undoubtedly reach lower performance.

The reminder of this paper is organized as follows.

In Section 2 some preliminaries on graph theory are

provided and the two main problems are stated. In

Section 3 we provide sufﬁcient conditions to solve a

leaderless and a leader-follower consensus problem,

and we give an LMI approach to tune the distributed

PID controller gains. We carry out simulations to test

the effectiveness of the proposed controller in Sec-

tion 4. The paper ends with conclusions and future

perspectives in Section 5.

2 PRELIMINARIES AND

PROBLEM STATEMENT

2.1 Graph Theory

An undirected graph G is a pair (V , E), where V =

{

1, . . . , N

}

is the set of nodes, and E ⊆ V × V is the

set of unordered pairs of nodes, named edges. Two

nodes i, j are said to be adjacent if (i, j) ∈ E. Un-

der the assumption of undirected graph, the latter im-

plies that ( j, i) ∈ E too. An undirected graph is con-

nected if there exists a path between every pair of dis-

tinct nodes, otherwise is disconnected. The adjacency

matrix A = [a

i j

] ∈ R

N×N

associated with the undi-

rected graph G, considered in this paper, is deﬁned by

= 0, i.e. self-loops are not allowed, and a

i j

= 1 if

(i, j) ∈ E. The Laplacian matrix L ∈ R

N×N

is deﬁned

as L

∑

j6=i

i j

and L

i j

= −a

i j

, i 6= j. Considering

an undirected graph we make use of the following

Lemma 1. (Ren et al., 2005) The Laplacian matrix

has the following properties: (i) L is symmetric and

all its eigenvalues are either strictly positive or equal

to 0, and 1 is the corresponding eigenvector to 0; (ii)

0 is a simple eigenvalue of L if and only if the graph

is connected.

We will also make use of another Laplacian matrix,

according to the following

Lemma 2. (Lin et al., 2008) Let

L =



i j



∈ R

N×N

a Laplacian matrix such that

i j

N − 1

if i = j, and

i j

= −

otherwise, then the following hold: (i) the

eigenvalues of

L are 1 with multiplicity N − 1, and

0 with multiplicity 1. 1

and 1 are respectively the

left and right eigenvector associated to eigenvalue 0;

(ii) there exists an orthogonal matrix U ∈ R

N×N

, i.e.

U : U

U = UU

= I, such that for any Laplacian

matrix L associated to any undirected graph we have

LU =



N−1

(N−1)×1

1×(N−1)



Λ,

LU =



(N−1)×1

1×(N−1)



A Distributed PID-like Consensus Control for Discrete-time Multi-agent Systems

where L

∈ R

(N−1)×(N−1)

is symmetric and positive

deﬁnite if the graph is connected.

In addition we employ the Kronecker product ⊗, for

which we have

Lemma 3. (Graham, 1981) Suppose that U ∈ R

p×p

V ∈ R

q×q

, X ∈ R

p×p

, and Y ∈ R

q×q

. The following

hold: (i) (U ⊗V ) (X ⊗Y ) = UX ⊗ VY ; (ii) suppose

U, and V invertible, then (U ⊗V )

−1

= U

−1

⊗V

−1

2.2 Problems Formulation

Problem 1. We consider N identical agents governed

by general discrete-time linear dynamics, according











= Ax

+ B

, i = 1, ·· · , N

= C

(1)

where A ∈ R

n×n

, B

∈ R

n×l

, B

∈ R

n×h

, C

∈ R

r×n

∈ R

m×n

, x

(k) ∈ R

and x

, x

(k + 1) ∈ R

are respectively the agent state at the current step k,

and at the next step k +1, u

, u

(k) ∈ R

is the agent

control, ω

, ω

(k) ∈ R

its disturbance, z

, z

(k) ∈

the variable on which agreement among the agents

is sought, and y

, y

(k) ∈ R

is the measured out-

put. For the sake of leaderless consensus, a priori

we do not require A to be Schur stable. Indeed, as

shown by (Ge et al., 2013), A has a role in determin-

ing the consensus function to which the agents con-

verge under proper control. Here it can be thought

to be assigned by a previous control design step. The

agents can communicate on an undirected connected

graph whose Laplacian matrix L has positive mini-

mum nonzero and maximum eigenvalues respectively

equal to λ

, and

. At this point, we can state the

problem in a general way as the one of ﬁnding a dis-

tributed control law for u

such that kz

− z

k is min-

imized for i, j = 1, ··· , N with respect to the distur-

bance ω , [ω

, ··· , ω

]

. In this work though, as

previously stated, we focus on local controllers of the

form

(

= A

+ B

, i = 1, ·· · , N

= C

+ D

(2)

where x

, x

(k) ∈ R

is the agent controller state,

and



l×l



2l×2l



− K

)



2l×m



l×l



l×2l

= [(K

+ K

)]

l×m

(3)

where K

, K

∈ R

l×m

are gain matrices to be

tuned, and where s

, s

(k) ∈ R

∑

j=1

i j

− y

) (4)

Thus the closed-loop system for agent i has dimen-

sion ¯n , n + 2l. As shown by (Wu et al., 2011), sys-

tem (2) is a state representation of the discrete-time

PID MIMO controller, whose z-transform is

(z)

= K

+ K

z − 1

+ K

z − 1

The problem can now be restated as the one of ﬁnding

matrices B

, and D

such that the effect of disturbance

ω on the consensus is minimized.

The second problem studied in this paper is the

following

Problem 2. Consider N + 1 discrete-time linear

agents, whose dynamics are described by











= Ax

+ B

= C











= Ax

+ B

, i = 1, ·· · , N

= C

(5)

where A ∈ R

n×n

, B

∈ R

n×h

, B

∈ R

n×l

, C

∈ R

r×n

∈ R

m×n

, x

(k) ∈ R

is the state of the N + 1

agent, called leader, y

, y

(k) ∈ R

is its mea-

sured output, u

, u

(k) ∈ R

is a time-varying un-

known control acting on the leader dynamics, and

, z

(k) ∈ R

is the variable on which we want

the follower controlled outputs z

to converge. Con-

cerning the remaining N follower agents, system de-

scription similar to (1) holds. The followers are as-

sumed to communicate on an undirected connected

graph whose Laplacian matrix is L. The leader can

pass information to a subset of followers. If agent

i receives information from the leader, then we set

to 1, and 0 otherwise. Thus we deﬁne M ,

L − diag(a

, ··· , a

), which is symmetric and pos-

itive deﬁnite, and we name λ

, and

respectively

its minimum and maximum eigenvalue. Without loss

of generality we consider A to be Schur stable. The

aim of the present problem is indeed not the one of sta-

bilizing each single agent, but rather to steer the fol-

lower agents state to the leader one despite the pres-

ence of u

, which makes the leader dynamics time-

varying. In order to accomplish such objective we aim

to employ the controller of form (2), (3), where we

consider a modiﬁed variable s

to take into account

ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics

the communication with the leader agent, according

∑

j=1

i j

− y

) + a

− y

) (6)

Intuitively such a controller is not capable of

solving the leader-follower tracking problem, i.e.

lim

k→∞

−z

k 6= 0 for i = 1, · · · , N, and for any vec-

tor signal u

, because the latter acts as an unknown

exogenous signal for the overall system including the

N + 1 agents. This is why we will focus on tuning the

controller matrices B

, and D

such that kz

− z

k is

minimized for i = 1, · · · , N.

3 MAIN RESULT

3.1 H

∞

Output Consensus

In order to state our main result we introduce the fol-

lowing deﬁnition, similar to the one given in (Wang

and Shao, 2015).

Deﬁnition 1. System (1) is said to achieve an H

∞

output consensus with a performance index γ ∈ R

if, for any initial condition, lim

k→∞

− z

k = 0 for

i, j = 1, ··· , N when ω = 0, and the H

∞

norms of the

transfer function matrices, for i = 1, ··· , N, between

ω and z

−

∑

j=1

are inferior to γ.

The following result is based on Theorem 3 in (Wu

et al., 2011), reported in Theorem 4 in the Appendix.

Theorem 1. Given N agents described by (1) on an

undirected connected graph; consider the distributed

protocol of equations (2),(3),(4); then the agents

achieve H

∞

output consensus with performance index

γ if there exist two symmetric positive deﬁnite matri-

ces P

P ∈ R

¯n× ¯n

such that the LMI conditions of The-

orem 4 are simultaneously satisﬁed for two LTI sys-

tems whose matrices are respectively (A, B

, λ

and (A, B

), and they both have controlled out-

put matrix C

, and disturbance input matrix B

Proof. The closed-loop dynamics for the generic

agent i, by using (1),(2), and by deﬁning the aug-

mented state ξ



, x



∈ R

¯n

, and matrices

m×2l

, [C

r×2l

B ,



h×(2l)



is given by

(

Aξ

∑

j=1

i j

(ξ

− ξ

) +

Bω

where

A =



A B

0 A



B =





(7)

Similar to (Liu et al., 2009), and (Wang and Gao,

2011), we deﬁne ζ

, z

−

∑

j=1

, and δ

, ξ

−

∑

j=1

, thus ζ

. Note that if ζ

= 0 for

i = 1, · · · , N then z

= z

, i.e. output consensus

is achieved. If now we name ξ ,



, ··· , ξ



δ ,



, ··· , δ



, and ζ ,



, ··· , ζ



, we have

that ζ =



⊗



δ, and δ = ξ − 1 ⊗

∑

j=1



L ⊗ I

¯n



ξ, where

L satisﬁes the conditions of

Lemma 2. Gathering together the equations of the

closed-loop agents dynamics, we obtain

(



⊗

A + L ⊗



ξ +



⊗



ζ =



⊗



L ⊗ I

¯n



ξ =



L ⊗



We now consider the following change of coordinates

δ =



L ⊗ I

¯n



ξ, which yields



L ⊗ I

¯n



⊗

A + L ⊗



ξ+



L ⊗ I

¯n



⊗



ω =



L ⊗

A +

LL ⊗



δ + 1 ⊗

∑

j=1



L ⊗





L ⊗

A +

LL ⊗



δ +



L ⊗



where we used points (i) of Lemma 2, and 3. Ac-

cording to the (ii) point of the former, we employ

the orthogonal matrix U ∈ R

N×N

to deﬁne the change

of coordinates:

δ ,



⊗ I

¯n



δ,

ω ,



⊗ I



ω,

ζ ,



⊗ I



ζ, so that the system equations in the

new coordinates are given by













⊗ I

¯n



L ⊗

A +

LL ⊗



(U ⊗ I

¯n

)



⊗ I

¯n



L ⊗





Λ ⊗

A +

ΛU

LU ⊗



δ +



Λ ⊗



ζ =



⊗ I



⊗



(U ⊗ I

¯n

)

δ =



⊗



(8)

As shown in Lemma 2, being the last rows of

Λ,

and U

LU zeros, we can split the dynamics (8) in

two parts by dividing the system variables as

δ =

[

]

ω = [

]

, and

ζ = [

]

. The

dynamic equation of the second variable is then

0, and it does not inﬂuences

. It follows that we can

study the reduced order system

(



N−1

⊗

A + L

⊗





N−1

⊗





N−1

⊗



From Lemma 2, it exists an orthogonal matrix V ∈

(N−1)×(N−1)

: V

V , Λ = diag(λ

, ··· , λ

N−1

A Distributed PID-like Consensus Control for Discrete-time Multi-agent Systems

where 0 < λ

≤ λ

≤

for i = 1, · · · , N − 1. Thus

we can deﬁne a further change of coordinates, such

that



⊗ I

¯n





⊗ I



, and



⊗ I



. The latter yields





N−1

⊗

A + Λ ⊗





N−1

⊗





N−1

⊗



(9)

It is easy to see that the transfer function matrix of (9)

satisﬁes

(z)k

∞

= kT

(z)k

∞

(z)k

∞

= kT

ζω

(z)k

∞

(10)

It follows that we can impose an H

∞

constraint on

transfer function matrix T

ζω

(z) by acting on T

(z).

We can now separate equation (9) in N − 1 subsys-

tems, each of them being governed by













(A + B

(λ

)) B

(λ

) A







= C

¯x

(11)

where

, [ ¯x

¯x

1,c

]

. System (11) can be equiva-

lently seen as the closed-loop form of the two follow-

ing systems











¯x

= A ¯x

+ B

¯u

+ B

¯y

, (λ

) ¯x

= C

¯x

(

¯x

1,c

= A

¯x

1,c

+ B

¯y

¯u

, C

¯x

1,c

+ D

¯y

(12)

Thus, we can reformulate the problem as the one ﬁnd-

ing matrices B

, and D

such that for i = 1, · · · , N − 1

the closed-loop system of (12) is Schur stable when

= 0, and to guarantee that kT

¯z

(z)k

∞

< γ. A sufﬁ-

cient condition to prove the existence of such a so-

lution and a relatively simple way to calculate the

controller matrices are obtained by employing The-

orem 4. In the latter it is proved that if it exists a sym-

metric positive deﬁnite matrix P

∈ R

¯n× ¯n

such that if

the given LMI conditions are satisﬁed, then closed-

loop system (11) using controller (2),(3),(4) is such

that

(k + 1)P

(k + 1) −

(k)P

(k)

< γ

(k)

(k) − ¯z

(k)¯z

(k)

It is important to stress that such LMI conditions are

afﬁne in the system matrices, variables and matrix P

We make use of this fact to provide sufﬁcient condi-

tions for which it exists a controller of the considered

form such that the mentioned LMI is simultaneously

veriﬁed for i = 1, · · · , N − 1. Since the generic eigen-

value of L

: λ

is such that λ

≤ λ

≤

, then it

always exists α

∈ R : 0 ≤ α

≤ 1 so that λ

= α

(1 − α

)

. Notice that the systems to be stabilized,

appearing in the ﬁrst set of equation in (12), can be

seen as one single system with an uncertain measure-

ment matrix, whose parameter is λ

. In other words,

, λ

, and ∃α

: C

= α

min

+ (1 − α

max

where C

min

, λ

, and C

mix

, i.e. it can

be written as a convex combination of the extreme

matrices C

min

, and C

max

. Thus, as in (Wang and

Gao, 2011), the proof makes use of classic results of

robust linear control, and in particular by introduc-

ing an afﬁne parameter dependent Lyapunov matrix

P(α

) , α

P + (1 − α

)

P, where P,

P are Lyapunov

matrices solution of simultaneous LMI of Theorem 4

written for respectively C

min

, and C

max

. Eventually,

it is easy to show that if P,

P exist, then the controller

solves the problem ∀λ ∈ R : λ

≤ λ ≤

, and in par-

ticular for λ = λ

, for i = 1, · · · , N − 1. Such a con-

troller is easily found from the solution of the afore-

mentioned LMI condition. Indeed among the LMI

variables there are matrices B

, and D

, from which

it is easy to calculate the PID gains K

, K

, and K

employing relations in (3).

Remark 1. Note that the mentioned LMI conditions,

if satisﬁed, guarantee that the consensus error is min-

imized with respect to the disturbance. However the

latter still have a role in determining the consensus

function to which the agents converge.

3.2 Leader-follower Consensus under

Time-varying Reference

The result of Subsection 3.1 can be easily adapted for

the sake of leader-follower consensus via an H

∞

for-

mulation of the problem. Thus, we give the following

Deﬁnition 2. System (5) is said to achieve an H

∞

out-

put leader-follower consensus with a performance in-

dex γ ∈ R

if, for any initial condition, lim

k→∞

−

k = 0 for i = 1, · · · , N when u

(k) = 0 ∀k ∈ N,

and the H

∞

norms of the transfer function matrices,

for i = 1, ··· , N, between u

and z

−z

are inferior to

γ.

Theorem 2. Given the system described by (5), where

N follower agents can communicate on an undirected

connected graph, and one leader can communicate

with a non-empty subset of followers; consider the

distributed protocol of equations (2),(3),(6); then the

systems achieve H

∞

output leader-follower consen-

sus with performance index γ if there exist two sym-

metric positive deﬁnite matrices P,

P ∈ R

¯n× ¯n

such

ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics

that the LMI conditions of Theorem 4 are simultane-

ously satisﬁed for two LTI systems whose matrices are

respectively (A, B

, λ

), and (A, B

), and

they both have controlled output matrix C

, and dis-

turbance input matrix −B

Proof. The proof is similar to proof of Subsection 3.1.

By deﬁning error e

, x

−x

, ξ , [e

, x

]

, and ζ

the closed-loop system for the generic follower

agent i is given by

(

Aξ



∑

j=1

i j

(ξ

− ξ

) + a



where

are deﬁned in (7), and

B ,

[−B

h×2l

]

. Deﬁning u

, 1 ⊗ u

, we then

gather the N agent equations together

(



⊗

A + M ⊗



ξ +



⊗



ζ =



⊗



(13)

From the deﬁnition of M in Section 2, there

exists a orthogonal matrix U : U

M U , Λ =

diag(λ

, ··· , λ

), where λ

∈ R : λ

> 0 for i =

1, ··· , N, so that we can deﬁne the change of coordi-

nates ξ , (U ⊗ I

¯n

)

ξ, u

, (U ⊗ I

)ˆu

, ζ , (U ⊗ I

)

ζ.

By applying similar calculation as in the previous sub-

section, the global system in the new coordinates

(



⊗

A + Λ ⊗



ξ +



⊗



ˆu

ζ =



⊗



(14)

As in (10), it results that kT

ξˆu

(z)k

∞

= kT

ξu

(z)k

∞

i.e. we can minimize the effect of u

on the con-

sensus error by acting on system (14). Similar to the

passage from equations (9) to (11), splitting (14) in N

subsystems yields the following equation for subsys-

tem i













(A + B

(λ

)) B

(λ

) A





−B



ˆu

= C

ˆe

(15)

where

, [ˆe

ˆx

]

. Equivalently, it can be de-

scribed as the connection of the two following sys-

tems











ˆe

= A ˆe

+ B

ˆu

− B

ˆu

ˆy

, (λ

) ˆe

= C

ˆe

(

ˆx

= A

ˆx

+ B

ˆy

ˆu

, C

ˆx

+ D

ˆy

(16)

The rest of the proof is equivalent to the last part

of Subsection 3.1, and it is concluded by invoking

Theorem 4, whose LMI conditions have to be simul-

taneously satisﬁed for the two systems at the ver-

tices of the polytope having matrices respectively

(A, B

, λ

), and (A, B

), and same con-

trolled output, and disturbance input matrices C

−B

. From the solution of the aforementioned LMIs

the controllers gains are easily found as in the proof of

Theorem 1. If such a solution exists, then the system

is stable.

Having employed a PID structure for the distributed

controller suggests that consensus should be reached

for any u

(k) = ¯u

, where ¯u

is any constant vector.

However this is not automatically guaranteed in the

MIMO case by the mentioned LMI conditions, and in

this framework it is veriﬁed a posteriori. Nonetheless,

if such LMI has a solution then, according to the well-

known Francis equation, a necessary condition for the

proposed controller to reject constant exogenous sig-

nals is that l ≥ r.

In the leader-follower consensus framework a dif-

ferent tuning of the PID controller gains with respect

to Theorem 2 could lead to better performance, as

shown in Section 4. Thus, by proposing the follow-

ing deﬁnition we aim to focus on system fast response

rather than imposing some H

∞

constraint. For this last

development, we further consider r = m, thus we sim-

ply name C , C

= C

Deﬁnition 3. System (5) is said to achieve fast leader-

follower consensus with performance index τ ∈ R

for u

(k) = 0, and any initial condition, lim

k→∞

−

k = 0 for i = 1, ··· , N, and (1−e

−1

)% of consensus

is achieved in a maximum number of steps equal to

Note that the same kind of deﬁnition can be consid-

ered for sampled-data systems, by saying that sys-

tem (5) achieves fast leader-follower consensus with

a time constant inferior to τT

, where T

is the system

sampling time. The result we present in the following

is based on Theorem 2 in (Wu et al., 2011), reported

in Theorem 5 in the Appendix.

Theorem 3. Given the system described by (5), where

N follower agents can communicate on an undi-

rected connected graph, and one leader can com-

municate with a non-empty subset of followers; con-

sider the distributed protocol of equations (2),(3),(6);

then the systems achieve fast leader-follower con-

sensus with performance index τ = −

log(R)

, where

R ∈ R : 0 ≤ R < 1, if there exist two symmetric pos-

itive deﬁnite matrices P,

P ∈ R

¯n× ¯n

such that the LMI

conditions of Theorem 5 are simultaneously satisﬁed

A Distributed PID-like Consensus Control for Discrete-time Multi-agent Systems

for two LTI systems whose matrices are respectively

(A, B

, λ

C), and (A, B

C), and where the real

constants (a, b) to be set in Theorem 5 are chosen to

be (a, b) = (0, R).

Proof. The proof employs the same change of co-

ordinates as in the previous one, so that we can re-

state the problem as the one of stabilizing the top

system of equation (16), for i = 1, · · · , N, with the

bottom system in (16), i.e. a PID controller whose

matrices are deﬁned in (3). Unlike Theorem 2, as

previously mentioned, we invoke Theorem 5, where

it is stated that given two real constants (a, b), if

there exists a symmetric positive deﬁnite matrix P

such that the given LMI conditions are satisﬁed, then

system (15) is stable with all its eigenvalues λ ly-

ing in the complex plane region deﬁned by F



(ℜ[λ], ℑ[λ]) : (ℜ[λ] + a)

+ ℑ[λ]

< b



. As for the

two previous proofs, we employ classic results of lin-

ear robust control to impose that this condition is

simultaneously satisﬁed for two systems at the ver-

tices of the polytope whose matrices are respectively

(A, B

, λ

C), and (A, B

C). If such a solution

exists then the eigenvalues of system (13) are guaran-

teed to lie in F

. In this framework we are interested

in speeding up the system response to u

. For this

reason we set a = 0, and b = R, where R : 0 ≤ R < 1.

Thus, all system eigenvalues are guaranteed to have a

module inferior to R. As a result, the system has the

slowest time-constant inferior to −

log(R)

. In terms

of number of iterations it is easy to see that such per-

formance is equal to a maximum value



−

log(R)



iterations. Eventually, from the LMI solution, the PID

gains are found as in the two previous proofs.

Remark 2. In this latter problem too, having imposed

a PID structure does not directly guarantee achieve-

ment of consensus for any constant u

(k) in the gen-

eral MIMO case. According to Francis equations, if

the mentioned LMI has a solution, a necessary condi-

tion though is given by l ≥ m.

4 SIMULATION EXAMPLES

First of all we carry out a numerical simulation to test

the H

∞

output consensus control. We consider a net-

work of 5 agents as show in Figure 1.a. Each of them

is governed by (1), where

A =

0.8182 0.0452 −0.0034

0 0.9888 −0.1492

0 0.1492 0.9888

1 0.4

0 1

0.5 0.5

, B

0.1

0.05

1.2 0.8 1.4

1.4 −1.2 0.8

−0.5 0.7 1.2

(17)

Note that (17) is not Schur stable because two of

its eigenvalues lay on the unit circle. Each agent

is perturbed by a disturbance of the form ω

(k) =

0.8ν

(k) + c

, where ν

is an aleatory variable with

uniform distribution of probability in [0, 1], and c

some constant value. The PID gains found via LMIs

allow an H

∞

performance index of γ = 0.18. Figure 2

shows the 5 agents trajectories (colored dashed lines)

as well as their average (blue continuous line). Then

we compare the two proposed PID gain tuning for a

leader-follower consensus problem. For this example

we consider the graph of Figure 1.b, where agent 0

is the leader. The system dynamics is governed by

equation (5), where

A =





0.7711 0.4744 0.2475

0.1646 0.4487 0.1036

−0.8959 −0.8534 −0.2198









0.5 0.3 0.4

0.7 0 1

0.4 0.9 0.3





, B





0.2

0.5

0.3







1 0 0



and C

as in (17). The controller tuned following The-

orem 2 allows an H

∞

performance index of γ = 2,

while the one tuned according to Theorem 3 guaran-

tees a performance index of τ = 6.1531. In Figure 3

we simulate the system step response for a value of

= 3. For ease of comparison, we plot here the

only output associated to matrix C

. As mentioned in

section 3, fast consensus (green dashed lines) outper-

forms the H

∞

one (red dashed-dotted lines). Indeed,

even if the latter respects Theorem 2, its consensus

error goes slowly to zero with respect to the former

one. Eventually, in Figure 4 it is shown the system

behavior for fast consensus tuning of PID gains when

is a time-varying vector signal, and where we set

C = C

= C



1 0 0

0 1 0



. The blue and the dark

green continuous signals are respectively leader states

, and x

, while the followers states are represented

respectively by the dashed green and red signals for

, and x

ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics

Figure 1: (a): leaderless communication graph (left);

(b): leader-follower communication graph (right).

Figure 2: H

∞

output consensus.

Figure 3: Step response comparison for H

∞

output and fast

leader-follower consensus.

Figure 4: Leader-follower consensus under time-varying

reference.

5 CONCLUSION

We presented a PID-like distributed protocol for gen-

eral LTI MIMO discrete-time agents communicating

on an undirected connected graph. By employing

LMIs we showed how the controller gains can be

tuned to solve two different, yet similar, problems,

namely a leaderless under system disturbances and

a leader-follower under time-varying reference state

consensus problem. Treating the system disturbances

in the H

∞

framework revealed good performance,

whereas a gain tuning based on fast response seems

to be preferable when dealing with a leader-follower

problem.

Our results are based on robust control to deal with

the problem of simultaneous stabilization of a given

number of systems. The given conditions are sufﬁ-

cient and therefore conservative. In near future work

we are interested in studying less restrictive condi-

tions when treating the discrete-time consensus prob-

lem in the H

∞

framework. Moreover, inspired by the

work of (Schiffer et al., 2016), we are currently study-

ing possible applications of the presented methods in

the engineering ﬁeld of power systems.

ACKNOWLEDGEMENTS

This study has been carried out in the RISEGrid In-

stitute (www.supelec.fr/342p38091/risegrid-en.html),

joint program between CentraleSup

elec and EDF

(’Electricit

e de France’) on smarter electric grids.

REFERENCES

Cao, Y., Ren, W., and Li, Y. (2009). Distributed discrete-

time coordinated tracking with a time-varying refer-

ence state and limited communication. Automatica,

45(5):1299–1305.

Carli, R., Chiuso, A., Schenato, L., and Zampieri, S.

(2008). A pi consensus controller for networked

clocks synchronization. IFAC Proceedings Volumes,

41(2):10289–10294.

Ge, Y., Chen, Y., Zhang, Y., and He, Z. (2013). State

consensus analysis and design for high-order discrete-

time linear multiagent systems. Mathematical Prob-

lems in Engineering, 2013.

Graham, A. (1981). Kronecker products and matrix calcu-

lus with applications. Holsted Press, New York.

Li, Z., Duan, Z., and Chen, G. (2011). On h

∞

and h

per-

formance regions of multi-agent systems. Automatica,

47(4):797–803.

Li, Z., Duan, Z., Xie, L., and Liu, X. (2012). Distributed

robust control of linear multi-agent systems with pa-

rameter uncertainties. International Journal of Con-

trol, 85(8):1039–1050.

Li, Z., Ren, W., Liu, X., and Fu, M. (2013). Distributed con-

tainment control of multi-agent systems with general

linear dynamics in the presence of multiple leaders.

International Journal of Robust and Nonlinear Con-

trol, 23(5):534–547.

Lin, P., Jia, Y., Du, J., and Yu, F. (2008). Distributed leadless

coordination for networks of second-order agents with

time-delay on switching topology. In 2008 American

Control Conference, pages 1564–1569. IEEE.

Liu, Y., Jia, Y., Du, J., and Yuan, S. (2009). Dynamic

output feedback control for consensus of multi-agent

systems: an h

∞

approach. In 2009 American Control

Conference, pages 4470–4475. IEEE.

Oh, K.-K., Moore, K. L., and Ahn, H.-S. (2014). Distur-

bance attenuation in a consensus network of identical

linear systems: An approach. IEEE Transactions on

Automatic Control, 59(8):2164–2169.

A Distributed PID-like Consensus Control for Discrete-time Multi-agent Systems

Ou, L.-L., Chen, J.-J., Zhang, D.-M., Zhang, L., and Zhang,

W.-D. (2014). Distributed h

∞

pid feedback for im-

proving consensus performance of arbitrary-delayed

multi-agent system. International Journal of Automa-

tion and Computing, 11(2):189–196.

Ren, W. (2007). Multi-vehicle consensus with a time-

varying reference state. Systems & Control Letters,

56(7):474–483.

Ren, W. and Beard, R. W. (2008). Distributed consensus in

multi-vehicle cooperative control. Springer.

Ren, W., Beard, R. W., et al. (2005). Consensus seeking

in multiagent systems under dynamically changing in-

teraction topologies. IEEE Transactions on automatic

control, 50(5):655–661.

Schiffer, J., Seel, T., Raisch, J., and Sezi, T. (2016). Voltage

stability and reactive power sharing in inverter-based

microgrids with consensus-based distributed voltage

control. IEEE Transactions on Control Systems Tech-

nology, 24(1):96–109.

Su, Y. and Huang, J. (2012). Two consensus problems for

discrete-time multi-agent systems with switching net-

work topology. Automatica, 48(9):1988–1997.

Wang, L. and Gao, L. (2011). H

∞

consensus control

for discrete-time multi-agent systems with switching

topology. Procedia Engineering, 15:601 – 607.

Wang, X. and Shao, J. (2015). Consensus for discrete-time

multiagent systems. Discrete Dynamics in Nature and

Society.

Wu, Z., Iqbal, A., and Amara, F. B. (2011). Lmi-based

multivariable pid controller design and its application

to the control of the surface shape of magnetic ﬂuid

deformable mirrors. IEEE Transactions on Control

Systems Technology, 19(4):717–729.

Xi, J., Cai, N., and Zhong, Y. (2010). Consensus prob-

lems for high-order linear time-invariant swarm sys-

tems. Physica A: Statistical Mechanics and its Appli-

cations, 389(24):5619–5627.

Xi, J., Shi, Z., and Zhong, Y. (2012). Output consen-

sus analysis and design for high-order linear swarm

systems: partial stability method. Automatica,

48(9):2335–2343.

Yang-Zhou, C., Yan-Rong, G., and ZHANG, Y.-X. (2014).

Partial stability approach to consensus problem of lin-

ear multi-agent systems. Acta Automatica Sinica,

40(11):2573–2584.

You, K. and Xie, L. (2011). Network topology and commu-

nication data rate for consensusability of discrete-time

multi-agent systems. IEEE Transactions on Automatic

Control, 56(10):2262–2275.

APPENDIX

In the following we report the two cited theorems of

(Wu et al., 2011). Consider the system of equations

(

= Ax + B

ω + B

z = C

x, y = C

(18)

where A ∈ R

n×n

, B

∈ R

n×l

, B

∈ R

n×h

, C

∈ R

r×n

∈ R

m×n

, x , x(k) ∈ R

and x

, x(k + 1) ∈ R

are respectively the system state at the current step

k, and at the next step k + 1, u , u(k) ∈ R

is the

control input, ω , ω(k) ∈ R

is an exogenous in-

put signal, z , z(k) ∈ R

the controlled output, and

y , y(k) ∈ R

is the measured one. Deﬁne the ma-

trices C



r×(2l)



B ,



h×(2l)



, K ,





, and

A ,



A B

2l×n



where A

, B

, C

, and D

are deﬁned in (3). Assum-

ing B

to be of full column rank without loss of gen-

erality, there exists an invertible T

∈ R

n×n

: T



l×(n−l)

l×l



. Finally deﬁne

T ,



n×2l

2l×n

2l×2l



Thus, we have the following theorems

Theorem 4. Consider system (18). If there exists a

positive deﬁnite matrix P ∈ R

¯n× ¯n

, where ¯n , n + 2l,

matrices

F =



( ¯n−q)×3l



∈ R

q×3l

, 1 ≤ q ≤ 3l, G

, [G

0] ∈ R

¯n× ¯n

, G

∈

¯n×( ¯n−3l)

, G

, [G

0] ∈ R

h× ¯n

, G

∈ R

h×( ¯n−3l)

, [G

0] ∈ R

r× ¯n

, G

∈ R

r×( ¯n−3l)

, H

∈ R

¯n×r

∈ R

¯n×r

, H

∈ R

h×r

, H

∈ R

r×r

, Y ∈ R

q×m

, and



( ¯n−q)×n

( ¯n−q)×2l

q×2l



and we further name Ψ

, P − FT − (FT )

, N

+ (FT

− G

T + (H

)

, Ψ

−P + G

A + (G

+ H

+ (H

)

, Ψ

(FT

− G

T , Ψ

, G

A + H

+ (G

, −γ

I +G

B+(G

, Ψ

, −G

T −H

, G

A + H

− H

, Ψ

, G

B − H

, and

, I − H

− H

, such that the following LMI has

a solution







∗ ∗ ∗

∗ ∗

∗







< 0

and if exists K such that F

K = Y , then the H

∞

norm

of the closed-loop system given by (18) and

(

= A

+ B

u = C

+ D

satisﬁes kT

zω

∞

< γ.

ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics

Theorem 5. Consider system (18). If there exists a

positive deﬁnite matrix P ∈ R

¯n× ¯n

, and a matrix

J =



( ¯n−q)×3l



∈ R

3l×3l

, and X ∈ R

3l×m

, and we further name

Ω ,



( ¯n−3l)×n

( ¯n−3l)×2l

3l×2l



such that the following LMI has a solution



bP ∗

Ω + JT

A + aJT b(JT + (JT )

− P)



> 0

and if J is nonsingular, then by choosing K = J

−1

the eigenvalues of the following matrix



(A + B

) B



lie in the region F



(ℜ[λ], ℑ[λ]) : (ℜ[λ] + a)

+ℑ[λ]

< b



A Distributed PID-like Consensus Control for Discrete-time Multi-agent Systems